3_homework3 - EE 211A Fall Quarter 2011 Instructor John...

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1 EE 211A Digital Image Processing I Fall Quarter, 2011 Handout 8 Instructor: John Villasenor Homework 3 Due: Thursday, 13 October 2011 Reading: Textbook pp. 20-31, 132-145. Textbook problems: 4.3 (The reconstruction filter should cut off at 0.5 0.5 , x y ± ± Δ Δ , not x Δ 5 . 0 , y Δ 5 . 0 as given in the book. Assume that frequencies exactly at the cutoff frequency are passed with no attenuation by the filter.), 4.9 1. Consider the following discrete functions: n n 1 h(m,n) = 1 -1 x(m,n) = 1 -1 2 0 1 1 -1 m m Each of the above sequences can be written as a matrix, where m is the row index and n is the column index. In this problem, express the 2D convolution ) , ( ) , ( ) , ( n m x n m h n m y = as a doubly Toeplitz block matrix H operating on a 4 by 1 column vector X obtained by column ordering the elements of the sequence ). , ( n m x (a) Give the vector . X (b) Specify the matrices , n H n = 0, 1, and 2, where n H are the non-zero blocks in the matrix . n H (c) Write the block matrix equation for the output , Y where Y is a column- ordered vector containing the elements of the output sequence ). , ( n m y Do not specify the elements of the block matrices, but be sure to label all blocks.
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